Exercise 572Logic, sets and proofs › Sets and mapsHigh schoolAdded 3 Nov 2026

Cardinality of finite sets

#cardinality#finite sets#inclusion-exclusion

Statement

Show that $|A\cup B|=|A|+|B|-|A\cap B|$ for finite sets.

Hint

Partition $A\cup B$ into $A\setminus B$ , $A\cap B$ , $B\setminus A$ .

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Partition. $A\cup B=(A\setminus B)\cup(A\cap B)\cup(B\setminus A)$ , disjoint.

Sizes. $|A|=|A\setminus B|+|A\cap B|$ , $|B|=|B\setminus A|+|A\cap B|$ .

Sum. $|A|+|B|=|A\cup B|+|A\cap B|$ , so $|A\cup B|=|A|+|B|-|A\cap B|$ .

Conclusion. The formula holds.